Abstract

The S-transform, conceptually viewed as an extension of the short-time Fourier transform and the wavelet transform, features a time-frequency representation known for its local spectral property and multiresolution strategy. It has been introduced in optical three-dimensional shape measurement based on a fringe projection technique recently. In this paper, an application of S-transform for demodulating fringe patterns affected by the nonlinearity has been discussed. Two methods based on the S-transform, called S-transform ridge method and S-transform filtering method, are used to eliminate the phase errors caused by nonlinear factors of the projector and CCD camera. The theoretical representations of S-transform ridge method and S-transform filtering method in fringe analysis are given. The computer simulation and experiment are carried out to verify our research. Compared with the reconstructions of the windowed Fourier transform, the wavelet transform and the S-transform ridge method, the S-transform filtering method gives a better result under considering the influence of the fringe nonlinearity.

© 2012 Optical Society of America

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2010

G. Mokryani, P. Siano, and A. Piccolo, “Identification of ferroresonance based on S-transform and support vector machine,” Simul. Model Pract. Theory 18, 1412–1424(2010).

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

2008

E. Sejdíc, I. Djurovíc, and J. Jiang, “A window width optimized S-transform,” EURASIP J. Advances Signal Processing 2008, 13–28 (2008).

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringed by using S-tranform,” Measurement 41, 403–411 (2008).
[CrossRef]

2007

S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktas, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett 32, 591–593 (2007).
[CrossRef]

J. G. Zhong and H. P. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46, 2670–2675 (2007).
[CrossRef]

2006

S. Z. Zheng, W. J. Chen, and X. Y. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
[CrossRef]

2005

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

2004

M. V. Chilukuri and P. K. Dash, “Multiresolution S-transform-based fuzzy recognition system for power quality events,” IEEE Trans. Power Delivery 19, 323–330 (2004).
[CrossRef]

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

J. G. Zhong and J. W. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

2002

2000

Y. I. Portnyagin, J. M. Forbes, E. G. Merzlyakov, N. A. Makarov, and S. E. Palo, “Intradiurnal wind variations observed in the lower thermosphere over the south pole,” Ann. Geophys. 18, 547–554 (2000).
[CrossRef]

1996

R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S-transform,” IEEE Trans. Signal Process. 44, 998–1001(1996).
[CrossRef]

1989

S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

1986

1985

1983

1982

1946

D. Gabor and Ing, “Theory of communications,” J. Inst. Elect. Eng. London 93, 429–457 (1946).

Beauville, F.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Bianchini, G.

Bizouard, M.-A.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Blackburn, L.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Bosi, L.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Brady, P.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Brocco, L.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Brown, D.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Brown, R. A.

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

Buskulic, D.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Chatterji, S.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Chen, W. J.

S. Z. Zheng, W. J. Chen, and X. Y. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
[CrossRef]

Chilukuri, M. V.

M. V. Chilukuri and P. K. Dash, “Multiresolution S-transform-based fuzzy recognition system for power quality events,” IEEE Trans. Power Delivery 19, 323–330 (2004).
[CrossRef]

Christensen, N.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Clapson, A.-C.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Cortesi, U.

Coskun, E.

S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktas, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett 32, 591–593 (2007).
[CrossRef]

Dash, P. K.

M. V. Chilukuri and P. K. Dash, “Multiresolution S-transform-based fuzzy recognition system for power quality events,” IEEE Trans. Power Delivery 19, 323–330 (2004).
[CrossRef]

Djurovíc, I.

E. Sejdíc, I. Djurovíc, and J. Jiang, “A window width optimized S-transform,” EURASIP J. Advances Signal Processing 2008, 13–28 (2008).

Dursun, A.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringed by using S-tranform,” Measurement 41, 403–411 (2008).
[CrossRef]

Ecevit, F. N.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringed by using S-tranform,” Measurement 41, 403–411 (2008).
[CrossRef]

Fairhurst, S.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Forbes, J. M.

Y. I. Portnyagin, J. M. Forbes, E. G. Merzlyakov, N. A. Makarov, and S. E. Palo, “Intradiurnal wind variations observed in the lower thermosphere over the south pole,” Ann. Geophys. 18, 547–554 (2000).
[CrossRef]

Gabor, D.

D. Gabor and Ing, “Theory of communications,” J. Inst. Elect. Eng. London 93, 429–457 (1946).

Göktas, H.

S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktas, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett 32, 591–593 (2007).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing Second Edition (Publishing House of Electronics Industry, 2006), pp. 80–84.

Grosjean, D.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Guelachvili, G.

Guidi, G.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Halioua, M.

Hello, P.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Ing,

D. Gabor and Ing, “Theory of communications,” J. Inst. Elect. Eng. London 93, 429–457 (1946).

Itoh, K.

Jiang, J.

E. Sejdíc, I. Djurovíc, and J. Jiang, “A window width optimized S-transform,” EURASIP J. Advances Signal Processing 2008, 13–28 (2008).

Katsavounidis, E.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Knight, M.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Kocahan, Ö.

S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktas, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett 32, 591–593 (2007).
[CrossRef]

Lauzon, M. L.

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

Law, A. G.

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

Lazzarini, A.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Lee, C.

Liu, H. C.

Lowe, R. P.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S-transform,” IEEE Trans. Signal Process. 44, 998–1001(1996).
[CrossRef]

Makarov, N. A.

Y. I. Portnyagin, J. M. Forbes, E. G. Merzlyakov, N. A. Makarov, and S. E. Palo, “Intradiurnal wind variations observed in the lower thermosphere over the south pole,” Ann. Geophys. 18, 547–554 (2000).
[CrossRef]

Mallat, S.

S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

Mansinha, L.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S-transform,” IEEE Trans. Signal Process. 44, 998–1001(1996).
[CrossRef]

Marion, F.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Merzlyakov, E. G.

Y. I. Portnyagin, J. M. Forbes, E. G. Merzlyakov, N. A. Makarov, and S. E. Palo, “Intradiurnal wind variations observed in the lower thermosphere over the south pole,” Ann. Geophys. 18, 547–554 (2000).
[CrossRef]

Mitchell, J. R.

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

Mokryani, G.

G. Mokryani, P. Siano, and A. Piccolo, “Identification of ferroresonance based on S-transform and support vector machine,” Simul. Model Pract. Theory 18, 1412–1424(2010).

Mours, B.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Mutoh, K.

Özder, S.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringed by using S-tranform,” Measurement 41, 403–411 (2008).
[CrossRef]

S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktas, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett 32, 591–593 (2007).
[CrossRef]

Palchetti, L.

Palo, S. E.

Y. I. Portnyagin, J. M. Forbes, E. G. Merzlyakov, N. A. Makarov, and S. E. Palo, “Intradiurnal wind variations observed in the lower thermosphere over the south pole,” Ann. Geophys. 18, 547–554 (2000).
[CrossRef]

Pascale, E.

Piccolo, A.

G. Mokryani, P. Siano, and A. Piccolo, “Identification of ferroresonance based on S-transform and support vector machine,” Simul. Model Pract. Theory 18, 1412–1424(2010).

Portnyagin, Y. I.

Y. I. Portnyagin, J. M. Forbes, E. G. Merzlyakov, N. A. Makarov, and S. E. Palo, “Intradiurnal wind variations observed in the lower thermosphere over the south pole,” Ann. Geophys. 18, 547–554 (2000).
[CrossRef]

Ricci, F.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Saraç, Z.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringed by using S-tranform,” Measurement 41, 403–411 (2008).
[CrossRef]

Sejdíc, E.

E. Sejdíc, I. Djurovíc, and J. Jiang, “A window width optimized S-transform,” EURASIP J. Advances Signal Processing 2008, 13–28 (2008).

Siano, P.

G. Mokryani, P. Siano, and A. Piccolo, “Identification of ferroresonance based on S-transform and support vector machine,” Simul. Model Pract. Theory 18, 1412–1424(2010).

Srinivasan, V.

Stockwell, R. G.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S-transform,” IEEE Trans. Signal Process. 44, 998–1001(1996).
[CrossRef]

Su, X. Y.

S. Z. Zheng, W. J. Chen, and X. Y. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
[CrossRef]

Takeda, M.

Topkara, H. S.

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringed by using S-tranform,” Measurement 41, 403–411 (2008).
[CrossRef]

Villanueva, R. J.

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

Villanueva-Oller, J.

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

Weng, J. W.

J. G. Zhong and J. W. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing Second Edition (Publishing House of Electronics Industry, 2006), pp. 80–84.

Zanolin, M.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

Zeng, H. P.

Zhang, S.

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

Zheng, S. Z.

S. Z. Zheng, W. J. Chen, and X. Y. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
[CrossRef]

Zhong, J. G.

J. G. Zhong and H. P. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46, 2670–2675 (2007).
[CrossRef]

J. G. Zhong and J. W. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Zhu, H.

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

Ann. Geophys.

Y. I. Portnyagin, J. M. Forbes, E. G. Merzlyakov, N. A. Makarov, and S. E. Palo, “Intradiurnal wind variations observed in the lower thermosphere over the south pole,” Ann. Geophys. 18, 547–554 (2000).
[CrossRef]

Appl. Opt.

Appl. Spectrosc.

Australian and New Zealand Indust. Appl. Math. J.

H. Zhu, R. A. Brown, R. J. Villanueva, J. Villanueva-Oller, M. L. Lauzon, J. R. Mitchell, and A. G. Law, “Progressive imaging: S-transform order,” Australian and New Zealand Indust. Appl. Math. J. 45, C1002–C1016 (2004).

Class. Quantum Grav.

F. Beauville, M.-A. Bizouard, L. Blackburn, L. Bosi, P. Brady, L. Brocco, D. Brown, D. Buskulic, S. Chatterji, N. Christensen, A.-C. Clapson, S. Fairhurst, D. Grosjean, G. Guidi, P. Hello, E. Katsavounidis, M. Knight, A. Lazzarini, F. Marion, B. Mours, F. Ricci, and M. Zanolin, “A first comparison of search methods for gravitational wave bursts using Ligo and Virgo simulated data,” Class. Quantum Grav. 22, S1293–S1301 (2005).
[CrossRef]

EURASIP J. Advances Signal Processing

E. Sejdíc, I. Djurovíc, and J. Jiang, “A window width optimized S-transform,” EURASIP J. Advances Signal Processing 2008, 13–28 (2008).

IEEE Trans. Pattern Anal. Machine Intell.

S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

IEEE Trans. Power Delivery

M. V. Chilukuri and P. K. Dash, “Multiresolution S-transform-based fuzzy recognition system for power quality events,” IEEE Trans. Power Delivery 19, 323–330 (2004).
[CrossRef]

IEEE Trans. Signal Process.

R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S-transform,” IEEE Trans. Signal Process. 44, 998–1001(1996).
[CrossRef]

J. Inst. Elect. Eng. London

D. Gabor and Ing, “Theory of communications,” J. Inst. Elect. Eng. London 93, 429–457 (1946).

Measurement

A. Dursun, Z. Saraç, H. S. Topkara, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringed by using S-tranform,” Measurement 41, 403–411 (2008).
[CrossRef]

Opt. Eng.

J. G. Zhong and J. W. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

S. Z. Zheng, W. J. Chen, and X. Y. Su, “Adaptive windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45, 063601 (2006).
[CrossRef]

Opt. Lasers Eng.

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

Opt. Lett

S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktas, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett 32, 591–593 (2007).
[CrossRef]

Simul. Model Pract. Theory

G. Mokryani, P. Siano, and A. Piccolo, “Identification of ferroresonance based on S-transform and support vector machine,” Simul. Model Pract. Theory 18, 1412–1424(2010).

Other

R. C. Gonzalez and R. E. Woods, Digital Image Processing Second Edition (Publishing House of Electronics Industry, 2006), pp. 80–84.

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Figures (9)

Fig. 1.
Fig. 1.

Scheme of the geometry.

Fig. 2.
Fig. 2.

Relationship between S-transform spectrum and Fourier transform spectrum. (a) 1D fringe pattern; (b) Fourier transform spectrum; (c) S-transform spectrum; (d) product by superimposing S spectrum along the red arrow; (e) 2D flattop filter.

Fig. 3.
Fig. 3.

Simulation: (a) simulated object; (b) deformed fringe pattern (white line stands for the 64th row).

Fig. 4.
Fig. 4.

Results based on FTP method. (a) Spectrum of the 64th row of the fringe pattern; (b) reconstruction surface; (c) error distribution.

Fig. 5.
Fig. 5.

Results based on wavelet transform. (a) Reconstruction surface; (b) error distribution.

Fig. 6.
Fig. 6.

Computer simulation based on spectrum of S filtering method. (a) Gray display of S spectrum of the 64th row of the fringe; (b) 3D display of S spectrum of the 64th row of the fringe; (c) spectrum obtained by adding the local S spectrum after filtering operation; (d) reconstruction surface; (e) error distribution.

Fig. 7.
Fig. 7.

Computer simulation based on S-transform ridge method. (a) Reconstruction surface; (b) error distribution.

Fig. 8.
Fig. 8.

Schematic of experimental equipment.

Fig. 9.
Fig. 9.

Experiment: (a) reference fringe; (b) 512th row of the reference fringe pattern; (c) deformed fringe pattern; (d) 512th row of the deformed fringe pattern; (e) spectrum of the 512th row of the fringe pattern; (f) fundamental spectrum of the 512th row of the fringe pattern by Fourier transform method; (g) reconstruction by Fourier transform method; (h) reconstruction by wavelet transform method; (i) reconstruction by S-transform ridge method; (j) fundamental spectrum of the 512th row of the fringe pattern by S-transform filtering method; (k) reconstruction by S-transform filtering method.

Equations (22)

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g(x,y)=a(x,y)+b(x,y)cos[2πf0+φ(x,y)].
g0(x,y)=a(x,y)+b(x,y)cos[2πf0+φ0(x,y)],
Δφ(x,y)2πf0dL0h(x,y).
S(τ,f)=+h(t)|f|2πexp[f2(τt)2]exp(i2πft)dt,
S(τ,f)=H(α+f)exp(2π2α2f2)exp(i2πατ)dα,
g(x,y)=[a(x,y)+b(x,y)cos(2πf0x+φ(x,y))]γ,
g=g+k1g+k2g2+k3g3,
g=C0+C1{exp[i(2πf0x+φ)]+exp[i(2πf0x+φ)]}+C2{exp[i2(2πf0x+φ)]+exp[i2(2πf0x+φ)]}+C3{exp[i3(2πf0x+φ)]+exp[i3(2πf0x+φ)]},
C0=a+k1a+k2(a2+12b2)+k3(a3+32ab2),C1=b2(1+k1+2k2a+3k3a2+32k3b2),C2=14b2(k2+3k3a),C3=18k3b3.
G(α)=C0(α)+C1(αf0)+C1*(α+f0)+C2(α2f0)+C2*(α+2f0)+C3(α3f0)+C3*(α+3f0),
S(τ,f)¯dτ=S(τ,f)F(τ,fτrid|fwlowfwhigh)dτ=C1(α+ff0)exp(2π2α2f2)exp(i2πατ)dαF(τ,fτrid|fwlowfwhigh)dτ=C1(α+ff0)exp(2π2α2f2)[exp(i2πατ)F(τ,fτrid|fwlowfwhigh)dτ]dα=C1(α+ff0)exp(2π2α2f2)δα,0dα=C1(ff0),
wei={0.5{1+cos([π(ffrid(τ)+fwlow/3)/(2fwlow/3)]}frid(τ)fwlowf<frid(τ)fwlow/31frid(τ)fwlow/3ffrid(τ)+fwhigh/30.5{1+cos([π(ffrid(τ)fwhigh/3)/(2fwhigh/3)]}frid(τ)+fwhigh/3<ffrid(τ)+fwhigh0else,
φ(x)=φ(τ)+φ(x)(xτ).
S(τ,f)=S0(τ,f)+S1(τ,f)+S1*(τ,f)+S2(τ,f)+S2*(τ,f)+S3(τ,f)+S3*(τ,f),
S0(τ,f)=C0exp(2πfτ)exp(2π2),S1(τ,f)=12C1exp{i[2π(f+f0)τ+φ(τ)]}exp{2π2f2[f+f0+φ(τ)2π]2},S1*(τ,f)=12C1exp{i[2π(f+f0)τ+φ(τ)]}exp{2π2f2[ff0φ(τ)2π]2},S2(τ,f)=12C2exp{i[2π(f+2f0)τ+2φ(τ)]}exp{2π2f2[f+2(f0+φ(τ)2π)]2},S2*(τ,f)=12C2exp{i[2π(f+2f0)τ+2φ(τ)]}exp{2π2f2[f2(f0+φ(τ)2π)]2},S3(τ,f)=12C3exp{i[2π(f+3f0)τ+3φ(τ)]}exp{2π2f2[f+3(f0+φ(τ)2π)]2},S3*(τ,f)=12C3exp{i[2π(f+3f0)τ+3φ(τ)]}exp{2π2f2[f3(f0+φ(τ)2π)]2}.
exp{2π2fτ12[fτ1f0φ(τ)2π]2}=exp{2π2fτ22[fτ22(f0φ(τ)2π)]2}=exp{2π2fτ32[fτ33(f0φ(τ)2π)]2}=exp(8π2)0.
S(τ,f)=S1(τ,f)+S2(τ,f)+S3(τ,f)=12C1exp{i[2π(f+f0)τ+φ(τ)]}exp{2π2f2[f+f0+φ(τ)2π]2}+12C2exp{i[2π(f+2f0)τ+2φ(τ)]}exp{2π2f2[f+2(f0+φ(τ)2π)]2}+12C3exp{i[2π(f+3f0)τ+3φ(τ)]}exp{2π2f2[f+3(f0+φ(τ)2π)]2}.
φS(τ,fτ1)=arctan{imag[S(τ,fτ1)]real[S(τ,fτ1)]}=2π(f+f0)τ+φ(τ),
φ(τ)=Ounwrap[ϕS(τ,fτ1)+2π(fτ1f0)τ].
Ire=[a+bcos(2πf0x)]γ+n(x,y),
Ide=[a+bcos(2πf0(x+tanθ·Z(x,y)))]γ+n(x,y),
Peaks(x,y)=3×(1x)2×exp[(x2)(y+1)2]10×(x5x3y5)×exp(x2y2)13×exp[(x+1)2y2].

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